Blow-ups of caloric measure in time varying domains and applications to two-phase problems
英文摘要：
We develop a method to study the structure of the common part of the boundaries of disjoint and possibly non-complementary time-varying domains in $\mathbb{R}^{n+1}$, $n \geq 2$, at the points of mutual absolute continuity of their respective caloric measures. Our set of techniques, which is based on parabolic tangent measures, allows us to tackle the following problems: 1) Let $Ω_1$ and $Ω_2$ be disjoint domains in $\mathbb{R}^{n+1}$, $n \geq 2$, which are quasi-regular for the heat equation and regular for the adjoint heat equation, and their complements satisfy a mild non-degeneracy hypothesis on the set $E$ of mutual absolute continuity of the associated caloric measures $ω_i$ with poles at $\bar{p}_i=(p_i,t_i)\inΩ_i$, $i=1,2$. Then, we obtain a parabolic analogue of the results of Kenig, Preiss, and Toro, i.e., we show that the parabolic Hausdorff dimension of $ω_1|_E$ is $n+1$ and the tangent measures of $ω_1$ at $ω_1$-a.e. point of $E$ are equal to a constant multiple of the parabolic $(n+1)$-Hausdorff measure restricted to hyperplanes containing a line parallel to the time-axis. 2) If, additionally, $ω_1$ and $ω_2$ are doubling, $\log \frac{dω_2|_E}{dω_1|_E} \in VMO(ω_1|_E)$, and $E$ is relatively open in the support of $ω_1$, then their tangent measures at {\it every} point of $E$ are caloric measures associated with adjoint caloric polynomials. As a corollary we obtain that in complementary $δ$-Reifenberg flat domains, if $δ$ is small enough and $\log \frac{dω_2}{dω_1} \in VMO(ω_1)$, then $Ω_1 \cap \{t<t_2\}$ is vanishing Reifenberg flat. This generalizes results of Kenig and Toro for the Laplacian. 3) We establish a parabolic version of a theorem of Tsirelson about triple-points for harmonic measure.
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